(a) Find the sum of all multiples of 9 lying between 300 and 700. OR (b) The $26^{th}$, $11^{th}$ and the last term of an A.P. are 0, 3 and -51 respectively. Find the common difference and the number of terms of the A.P.

Step 1: Formulate equations from given terms

We are given the 26th term ($a_{26}$) and the 11th term ($a_{11}$) of an Arithmetic Progression (A.P.). The general formula for the $n^{th}$ term of an A.P. is $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. Using this formula, we can set up two linear equations.

Step 2: Solve for common difference (d)

To find the common difference $d$, we can subtract the second equation from the first. This eliminates the variable $a$, allowing us to solve for $d$.

Step 3: Solve for the first term (a)

Now that we have the value of $d$, we can substitute it back into either of the original equations to find the first term $a$. We'll use the equation $a + 10d = 3$.

Step 4: Find the number of terms (n)

We are given that the last term of the A.P. is $-51$. Using the $n^{th}$ term formula again, $a_n = a + (n-1)d$, we can substitute the values of $a$, $d$, and $a_n$ to solve for $n$, the total number of terms.

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